17 research outputs found
An equivalent formulation of the Fan-Raspaud Conjecture and related problems
In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture originally proposed by the first author, which states that in every simple bridgeless cubic graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a variant of Petersen-colourings, we prove it for graphs having oddness at most four and we give a natural extension to bridgeless cubic multigraphs and to certain cubic graphs having bridges
Circuits, Perfect Matchings and Paths in Graphs
We primarily consider the problem of finding a family of circuits to cover a bidgeless graph (mainly on cubic graph) with respect to a given weight function defined on the edge set. The first chapter of this thesis is going to cover all basic concepts and notations will be used and a survey of this topic.;In Chapter two, we shall pay our attention to the Strong Circuit Double Cover Conjecture (SCDC Conjecture). This conjecture was verified for some graphs with special structure. As the complement of two factor in cubic graph, the Berge-Fulkersen Conjecture was introduced right after SCDC Conjecture. In Chapter three, we shall present a series of conjectures related to perfect matching covering and point out their relationship.;In last chapter, we shall introduce the saturation number, in contrast to extremal number (or known as Turan Number), and describe the edge spectrum of saturation number for small paths, where the spectrum was consisted of all possible integers between saturation number and Turan number
REDUCTION OF THE BERGE-FULKERSON CONJECTURE TO CYCLICALLY 5-EDGE-CONNECTED SNARKS
The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic (3-valent) graph can be covered with six perfect matchings in such a way that every edge is in exactly two of them. As with several other classical conjectures in graph theory, every counterexample to the Berge-Fulkerson conjecture must be a non-3-edge-colorable cubic graph. In contrast to Tutte's 5-flow conjecture and the cycle double conjecture, no nontrivial reduction is known for the Berge-Fulkerson conjecture. In the present paper, we prove that a possible minimum counterexample to the conjecture must be cyclically 5-edge-connected
Some snarks are worse than others
Many conjectures and open problems in graph theory can either be reduced to
cubic graphs or are directly stated for cubic graphs. Furthermore, it is known
that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless
cubic graph which is not 3--edge-colourable. In this paper we deal with the
fact that the family of potential counterexamples to many interesting
conjectures can be narrowed even further to the family of
bridgeless cubic graphs whose edge set cannot be covered with four perfect
matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover
Conjecture and the Fan-Raspaud Conjecture are examples of statements for which
is crucial. In this paper, we study parameters which have
the potential to further refine and thus enlarge the set of
cubic graphs for which the mentioned conjectures can be verified. We show that
can be naturally decomposed into subsets with increasing
complexity, thereby producing a natural scale for proving these conjectures.
More precisely, we consider the following parameters and questions: given a
bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii)
how many copies of the same perfect matching need to be added, and (iii) how
many 2--factors need to be added so that the resulting regular graph is Class
I? We present new results for these parameters and we also establish some
strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure
On Fulkerson conjecture
If is a bridgeless cubic graph, Fulkerson conjectured that we can find 6
perfect matchings (a{\em Fulkerson covering}) with the property that every edge
of is contained in exactly two of them. A consequence of the Fulkerson
conjecture would be that every bridgeless cubic graph has 3 perfect matchings
with empty intersection (this problem is known as the Fan Raspaud Conjecture).
A {\em FR-triple} is a set of 3 such perfect matchings. We show here how to
derive a Fulkerson covering from two FR-triples. Moreover, we give a simple
proof that the Fulkerson conjecture holds true for some classes of well known
snarks.Comment: Accepted for publication in Discussiones Mathematicae Graph Theory;
Discussiones Mathematicae Graph Theory (2010) xxx-yy